Viscoelastic control of spatiotemporal order in bacterial active matter.

Active matter consists of units that generate mechanical work by consuming energy1. Examples include living systems (such as assemblies of bacteria2-5 and biological tissues6,7), biopolymers driven by molecular motors8-11 and suspensions of synthetic self-propelled particles12-14. A central goal is to understand and control the self-organization of active assemblies in space and time. Most active systems exhibit either spatial order mediated by interactions that coordinate the spatial structure and the motion of active agents12,14,15 or the temporal synchronization of individual oscillatory dynamics2. The simultaneous control of spatial and temporal organization is more challenging and generally requires complex interactions, such as reaction-diffusion hierarchies16 or genetically engineered cellular circuits2. Here we report a simple technique to simultaneously control the spatial and temporal self-organization of bacterial active matter. We confine dense active suspensions of Escherichia coli cells and manipulate a single macroscopic parameter-namely, the viscoelasticity of the suspending fluid- through the addition of purified genomic DNA. This reveals self-driven spatial and temporal organization in the form of a millimetre-scale rotating vortex with periodically oscillating global chirality of tunable frequency, reminiscent of a torsional pendulum. By combining experiments with an active-matter model, we explain this behaviour in terms of the interplay between active forcing and viscoelastic stress relaxation. Our findings provide insight into the influence of bacterial motile behaviour in complex fluids, which may be of interest in health- and ecology-related research, and demonstrate experimentally that rheological properties can be harnessed to control active-matter flows17,18. We envisage that our millimetre-scale, tunable, self-oscillating bacterial vortex may be coupled to actuation systems to act a 'clock generator' capable of providing timing signals for rhythmic locomotion of soft robots and for programmed microfluidic pumping19, for example, by triggering the action of a shift register in soft-robotic logic devices20.

Design rules for controlling active topological defects.

Topological defects play a central role in the physics of many materials, including magnets, superconductors, and liquid crystals. In active fluids, defects become autonomous particles that spontaneously propel from internal active stresses and drive chaotic flows stirring the fluid. The intimate connection between defect textures and active flow suggests that properties of active materials can be engineered by controlling defects, but design principles for their spatiotemporal control remain elusive. Here, we propose a symmetry-based additive strategy for using elementary activity patterns, as active topological tweezers, to create, move, and braid such defects. By combining theory and simulations, we demonstrate how, at the collective level, spatial activity gradients act like electric fields which, when strong enough, induce an inverted topological polarization of defects, akin to a negative susceptibility dielectric. We harness this feature in a dynamic setting to collectively pattern and transport interacting active defects. Our work establishes an additive framework to sculpt flows and manipulate active defects in both space and time, paving the way to design programmable active and living materials for transport, memory, and logic.

Patterning of morphogenetic anisotropy fields.

Orientational order, encoded in anisotropic fields, plays an important role during the development of an organism. A striking example of this is the freshwater polyp Hydra, where topological defects in the muscle fiber orientation have been shown to localize to key features of the body plan. This body plan is organized by morphogen concentration gradients, raising the question how muscle fiber orientation, morphogen gradients and body shape interact. Here, we introduce a minimal model that couples nematic orientational order to the gradient of a morphogen field. We show that on a planar surface, alignment to a radial concentration gradient can induce unbinding of topological defects, as observed during budding and tentacle formation in Hydra, and stabilize aster/vortex-like defects, as observed at a Hydra's mouth. On curved surfaces mimicking the morphologies of Hydra in various stages of development-from spheroid to adult-our model reproduces the experimentally observed reorganization of orientational order. Our results suggest how gradient alignment and curvature effects may work together to control orientational order during development and lay the foundations for future modeling efforts that will include the tissue mechanics that drive shape deformations.

Emergence of dynamic vortex glasses in disordered polar active fluids.

In equilibrium, disorder conspires with topological defects to redefine the ordered states of matter in systems as diverse as crystals, superconductors, and liquid crystals. Far from equilibrium, however, the consequences of quenched disorder on active condensed matter remain virtually uncharted. Here, we reveal a state of strongly disordered active matter with no counterparts in equilibrium: a dynamical vortex glass. Combining microfluidic experiments and theory, we show how colloidal flocks collectively cruise through disordered environments without relaxing the topological singularities of their flows. The resulting state is highly dynamical but the flow patterns, shaped by a finite density of frozen vortices, are stationary and exponentially degenerated. Quenched isotropic disorder acts as a random gauge field turning active liquids into dynamical vortex glasses. We argue that this robust mechanism should shape the collective dynamics of a broad class of disordered active matter, from synthetic active nematics to collections of living cells exploring heterogeneous media.

Vorticity phase separation and defect lattices in the isotropic phase of active liquid crystals.

We use numerical simulations and linear stability analysis to study the dynamics of an active liquid crystal film on a substrate in the regime where the passive system would be isotropic. Extensile activity builds up local orientational order and destabilizes the quiescent isotropic state above a critical activity, eventually resulting in spatiotemporal chaotic dynamics akin to the one observed ubiquitously in the nematic state. Here we show that tuning substrate friction yields a variety of emergent structures at intermediate activity, including lattices of flow vortices with associated regular arrangements of topological defects and a new state where flow vortices trap pairs of +1/2 defect that chase each other's tail. These chiral units spontaneously pick the sense of rotation and organize in a hexagonal lattice, surrounded by a diffuse flow of opposite rotation to maintain zero net vorticity. The length scale of these emergent structures is set by the screening length of the flow, controlled by the shear viscosity η and the substrate friction Γ, and can be captured by simple mode selection of the vortical flows. We demonstrate that the emergence of coherent structures can be interpreted as a phase separation of vorticity, where friction plays a role akin to that of birth/death processes in breaking conservation of the phase separating species and selecting a characteristic scale for the patterns. Our work shows that friction provides an experimentally accessible tuning parameter for designing controlled active flows.

Finite elasticity of the vertex model and its role in rigidity of curved cellular tissues.

Using a mean field approach and simulations, we study the non-linear mechanical response of the vertex model (VM) of biological tissue to compression and dilation. The VM is known to exhibit a transition between solid and fluid-like, or floppy, states driven by geometric incompatibility. Target perimeter and area set a target shape which may not be geometrically achievable, thereby engendering frustration. Previously, an asymmetry in the linear elastic response was identified at the rigidity transition between compression and dilation. Here we show that the asymmetry extends away from the transition point for finite strains. Under finite compression, an initially solid VM can completely relax perimeter tension, resulting in a drop discontinuity in the mechanical response. Conversely, an initially floppy VM under dilation can rigidify and have a higher response. These observations imply that re-scaling of cell area shifts the transition between rigid and floppy states. Based on this insight, we calculate the re-scaling of cell area engendered by intrinsic curvature and write a prediction for the rigidity transition in the presence of curvature. The shift of the rigidity transition in the presence of curvature for the VM provides a new metric for predicting tissue rigidity from image data of curved tissues in a manner analogous to the flat case.

Motility induced phase separation of deformable cells.

Using a multi-phase field model, we examine how particle deformability, which is a proxy for cell stiffness, affects motility induced phase separation (MIPS). We show that purely repulsive deformable, i.e., squishy, cells phase separate more effectively than their rigid counterparts. This can be understood as due to the fact that deformability increases the effective duration of collisions. In addition, the dense regions become increasingly disordered as deformability increases. Our results contextualize the applicability of MIPS to biological systems and have implications for how cells in biological systems may self-organize.

The role of non-affine deformations in the elastic behavior of the cellular vertex model.

The vertex model of epithelia describes the apical surface of a tissue as a tiling of polygonal cells, with a mechanical energy governed by deviations in cell shape from preferred, or target, area, A0, and perimeter, P0. The model exhibits a rigidity transition driven by geometric incompatibility as tuned by the target shape index, . For with p*(6) the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and preferred perimeter. As a result, the tissue is in a mechanically soft compatible state, with zero shear and Young's moduli. For p0 < p*(6), it is geometrically impossible for any cell to realize the preferred area and perimeter simultaneously, and the tissue is in an incompatible rigid solid state. Using a mean-field approach, we present a complete analytical calculation of the linear elastic moduli of an ordered vertex model. We analyze a relaxation step that includes non-affine deformations, leading to a softer response than previously reported. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape. The bulk modulus exhibits a jump discontinuity at the transition and can be lower in the rigid state than in the fluid-like state. The Poisson's ratio can become negative which lowers the bulk and Young's moduli. Our work provides a unified treatment of linear elasticity for the vertex model and demonstrates that this linear response is protocol-dependent.

Nonreciprocity as a generic route to traveling states.

We examine a nonreciprocally coupled dynamical model of a mixture of two diffusing species. We demonstrate that nonreciprocity, which is encoded in the model via antagonistic cross-diffusivities, provides a generic mechanism for the emergence of traveling patterns in purely diffusive systems with conservative dynamics. In the absence of nonreciprocity, the binary fluid mixture undergoes a phase transition from a homogeneous mixed state to a demixed state with spatially separated regions rich in one of the two components. Above a critical value of the parameter tuning nonreciprocity, the static demixed pattern acquires a finite velocity, resulting in a state that breaks both spatial and time-reversal symmetry, as well as the reflection parity of the static pattern. We elucidate the generic nature of the transition to traveling patterns using a minimal model that can be studied analytically. Our work has direct relevance to nonequilibrium assembly in mixtures of chemically interacting colloids that are known to exhibit nonreciprocal effective interactions, as well as to mixtures of active and passive agents where traveling states of the type predicted here have been observed in simulations. It also provides insight on transitions to traveling and oscillatory states seen in a broad range of nonreciprocal systems with nonconservative dynamics, from reaction-diffusion and prey-predators models to multispecies mixtures of microorganisms with antagonistic interactions.

Activity-Suppressed Phase Separation.

We use a continuum model to examine the effect of activity on a phase-separating mixture of an extensile active nematic and a passive fluid. We highlight the distinct role of (i) previously considered interfacial active stresses and (ii) bulk active stresses that couple to liquid crystalline degrees of freedom. Interfacial active stresses can arrest phase separation, as previously demonstrated. Bulk extensile active stresses can additionally strongly suppress phase separation by sustained self-stirring of the fluid, substantially reducing the size of the coexistence region in the temperature-concentration plane relative to that of the passive system. The phase-separated state is a dynamical emulsion of continuously splitting and merging droplets, as suggested by recent experiments. Using scaling analysis and simulations, we identify various regimes for the dependence of droplet size on activity. These results can provide a criterion for identifying the mechanisms responsible for arresting phase separation in experiments.

Shear-Driven Solidification and Nonlinear Elasticity in Epithelial Tissues.

Biological processes, from morphogenesis to tumor invasion, spontaneously generate shear stresses inside living tissue. The mechanisms that govern the transmission of mechanical forces in epithelia and the collective response of the tissue to bulk shear deformations remain, however, poorly understood. Using a minimal cell-based computational model, we investigate the constitutive relation of confluent tissues under simple shear deformation. We show that an initially undeformed fluidlike tissue acquires finite rigidity above a critical applied strain. This is akin to the shear-driven rigidity observed in other soft matter systems. Interestingly, shear-driven rigidity can be understood by a critical scaling analysis in the vicinity of the second order critical point that governs the liquid-solid transition of the undeformed system. We further show that a solidlike tissue responds linearly only to small strains and but then switches to a nonlinear response at larger stains, with substantial stiffening. Finally, we propose a mean-field formulation for cells under shear that offers a simple physical explanation of shear-driven rigidity and nonlinear response in a tissue.

Local Yield and Compliance in Active Cell Monolayers.

The rheology of biological tissue plays an important role in many processes, from organ formation to cancer invasion. Here, we use a multiphase field model of motile cells to simulate active microrheology within a tissue monolayer. When unperturbed, the tissue exhibits a transition between a solidlike state and a fluidlike state tuned by cell motility and deformability-the ratio of the energetic costs of steric cell-cell repulsion and cell-edge tension. When perturbed, solid tissues exhibit local yield-stress behavior, with a threshold force for the onset of motion of a probe particle that vanishes upon approaching the solid-to-liquid transition. This onset of motion is qualitatively different in the low and high deformability regimes. At high deformability, the tissue is amorphous when solid, it responds compliantly to deformations, and the probe transition to motion is smooth. At low deformability, the monolayer is more ordered translationally and stiffer, and the onset of motion appears discontinuous. Our results suggest that cellular or nanoparticle transport in different types of tissues can be fundamentally different and point to ways in which it can be controlled.

Fluctuations can induce local nematic order and extensile stress in monolayers of motile cells.

Recent experiments in various cell types have shown that two-dimensional tissues often display local nematic order, with evidence of extensile stresses manifest in the dynamics of topological defects. Using a mesoscopic model where tissue flow is generated by fluctuating traction forces coupled to the nematic order parameter, we show that the resulting tissue dynamics can spontaneously produce local nematic order and an extensile internal stress. A key element of the model is the assumption that in the presence of local nematic alignment, cells preferentially crawl along the nematic axis, resulting in anisotropy of fluctuations. Our work shows that activity can drive either extensile or contractile stresses in tissue, depending on the relative strength of the contractility of the cortical cytoskeleton and tractions by cells on the extracellular matrix.

A nonequilibrium force can stabilize 2D active nematics.

Suspensions of actively driven anisotropic objects exhibit distinctively nonequilibrium behaviors, and current theories predict that they are incapable of sustaining orientational order at high activity. By contrast, here we show that nematic suspensions on a substrate can display order at arbitrarily high activity due to a previously unreported, potentially stabilizing active force. This force moreover emerges inevitably in theories of active orientable fluids under geometric confinement. The resulting nonequilibrium ordered phase displays robust giant number fluctuations that cannot be suppressed even by an incompressible solvent. Our results apply to virtually all experimental assays used to investigate the active nematic ordering of self-propelled colloids, bacterial suspensions, and the cytoskeleton and have testable implications in interpreting their nonequilibrium behaviors.

Solid-Liquid Transition of Deformable and Overlapping Active Particles.

Experiments and theory have shown that cell monolayers and epithelial tissues exhibit solid-liquid and glass-liquid transitions. These transitions are biologically relevant to our understanding of embryonic development, wound healing, and cancer. Current models of confluent epithelia have focused on the role of cell shape, with less attention paid to cell extrusion, which is key for maintaining homeostasis in biological tissue. Here, we use a multiphase field model to study the solid-liquid transition in a confluent monolayer of deformable cells. Cell overlap is allowed and provides a way for modeling the precursor for extrusion. When cells overlap rather than deform, we find that the melting transition changes from continuous to first order like, and that there is an intermittent regime close to the transition, where solid and liquid states alternate over time. By studying the dynamics of five- and sevenfold disclinations in the hexagonal lattice formed by the cell centers, we observe that these correlate with spatial fluctuations in the cellular overlap, and that cell extrusion tends to initiate near fivefold disclinations.

Swimmer Suspensions on Substrates: Anomalous Stability and Long-Range Order.

We present a comprehensive theory of the dynamics and fluctuations of a two-dimensional suspension of polar active particles in an incompressible fluid confined to a substrate. We show that, depending on the sign of a single parameter, a state with polar orientational order is anomalously stable (or anomalously unstable), with a nonzero relaxation (or growth) rate for angular fluctuations, not parallel to the ordering direction, at zero wave number. This screening of the broken-symmetry mode in the stable state does lead to conventional rather than giant number fluctuations as argued by Bricard et al., Nature 503, 95 (2013), but their bend instability in a splay-stable flock does not exist and the polar phase has long-range order in two dimensions. Our theory also describes confined three-dimensional thin-film suspensions of active polar particles as well as dense compressible active polar rods, and predicts a flocking transition without a banding instability.

Information and motility exchange in collectives of active particles.

We examine the interplay of motility and information exchange in a model of run-and-tumble active particles where the particle's motility is encoded as a bit of information that can be exchanged upon contact according to the rules of AND and OR logic gates in a circuit. Motile AND particles become non-motile upon contact with a non-motile particle. Conversely, motile OR particles remain motile upon collision with their non-motile counterparts. AND particles that have become non-motile additionally "reawaken", i.e., recover their motility, at a fixed rate µ, as in the SIS (susceptible, infected, susceptible) model of epidemic spreading, where an infected agent can become healthy again, but keeps no memory of the recent infection, hence it is susceptible to a renewed infection. For µ = 0, both AND and OR particles relax irreversibly to absorbing states of all non-motile or all motile particles, respectively. The relaxation kinetics is, however, faster for OR particles that remain active throughout the process. At finite µ, the AND dynamics is controlled by the interplay between reawakening and collision rates. The system evolves to a state of all motile particles (an absorbing state in the language of absorbing phase transitions) for µ > µc and to a mixed state with coexisting motile and non-motile particles (an active state in the language of absorbing phase transitions) for µ < µc. The final state exhibits a rich structure controlled by motility-induced aggregation. Our work can be relevant to biochemical signaling in motile bacteria, the spreading of epidemics and of social consensus, as well as light-controlled organization of active colloids.

Shape and size changes of adherent elastic epithelia.

Epithelial tissues play a fundamental role in various morphogenetic events during development and early embryogenesis. Although epithelial monolayers are often modeled as two-dimensional (2D) elastic surfaces, they distinguish themselves from conventional thin elastic plates in three important ways- the presence of an apical-basal polarity, spatial variability of cellular thickness, and their nonequilibrium active nature. Here, we develop a minimal continuum model of a planar epithelial tissue as an active elastic material that incorporates all these features. We start from a full three-dimensional (3D) description of the tissue and derive an effective 2D model that captures, through the curvature of the apical surface, both the apical-basal asymmetry and the spatial geometry of the tissue. Crucially, variations of active stresses across the apical-basal axis lead to active torques that can drive curvature transitions. By identifying four distinct sources of activity, we find that bulk active stresses arising from actomyosin contractility and growth compete with boundary active tensions due to localized actomyosin cables and lamellipodial activity to generate the various states spanning the morphospace of a planar epithelium. Our treatment hence unifies 3D shape deformations through the coupled mechanics of apical curvature change and in-plane expansion/contraction of substrate-adhered tissues. Finally, we discuss the implications of our results for some biologically relevant processes such as tissue folding at the onset of lumen formation.

Small-scale demixing in confluent biological tissues.

Surface tension governed by differential adhesion can drive fluid particle mixtures to sort into separate regions, i.e., demix. Does the same phenomenon occur in confluent biological tissues? We begin to answer this question for epithelial monolayers with a combination of theory via a vertex model and experiments on keratinocyte monolayers. Vertex models are distinct from particle models in that the interactions between the cells are shape-based, as opposed to distance-dependent. We investigate whether a disparity in cell shape or size alone is sufficient to drive demixing in bidisperse vertex model fluid mixtures. Surprisingly, we observe that both types of bidisperse systems robustly mix on large lengthscales. On the other hand, shape disparity generates slight demixing over a few cell diameters, a phenomenon we term micro-demixing. This result can be understood by examining the differential energy barriers for neighbor exchanges (T1 transitions). Experiments with mixtures of wild-type and E-cadherin-deficient keratinocytes on a substrate are consistent with the predicted phenomenon of micro-demixing, which biology may exploit to create subtle patterning. The robustness of mixing at large scales, however, suggests that despite some differences in cell shape and size, progenitor cells can readily mix throughout a developing tissue until acquiring means of recognizing cells of different types.

Correlating cell shape and cellular stress in motile confluent tissues.

Collective cell migration is a highly regulated process involved in wound healing, cancer metastasis, and morphogenesis. Mechanical interactions among cells provide an important regulatory mechanism to coordinate such collective motion. Using a self-propelled Voronoi (SPV) model that links cell mechanics to cell shape and cell motility, we formulate a generalized mechanical inference method to obtain the spatiotemporal distribution of cellular stresses from measured traction forces in motile tissues and show that such traction-based stresses match those calculated from instantaneous cell shapes. We additionally use stress information to characterize the rheological properties of the tissue. We identify a motility-induced swim stress that adds to the interaction stress to determine the global contractility or extensibility of epithelia. We further show that the temporal correlation of the interaction shear stress determines an effective viscosity of the tissue that diverges at the liquid-solid transition, suggesting the possibility of extracting rheological information directly from traction data.